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A NEW PRIMALITY CRITERION OF MANN AND SHANKS AND ITS RELATION TO A THEOREM OF HERMITE WITH EXTENSION TO FIBONOMIALS H.W.GOULD W. Virginia University, Morgantown, West Virginia 1. INTRODUCTION Henry Mann and Daniel Shanks [4] have found a new n e c e s s a r y and sufficient condition for a number to be a p r i m e . This criterion may be stated in novel t e r m s as a property of a displaced Pascal Arithmetical Triangle as follows* Consider a rectangular a r r a y of numbers made by moving each row in the usual Pascal a r r a y two places to the right from the previous row. The n + 1 binomial coefficients f M , k = 0, 1, •• • , n, row between columns 2n and 3n inclusive. a r e then found in the nth We shall say that a given column has the Row Divisibility Property if each entry in the column is divisible by the corresponding row number. Then the new criterion is that the column number is a prime if and only if the column has the Row Divisibility Property. Column Number 0 0 1 2 3 1 1 1 4 5 6 1 2 1 7 8 9 3 3 1 13 14 15 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 1 8 28 56 1 9 10 11 12 6 4 1 1 5 16 17 18 19 1 2 3 4 5 1 1 4 6 7 8 9 Row Number The Displaced A r r a y We wish to show h e r e that by relabelling the a r r a y it is easy to relate the new criterion with a theorem of Hermite on factors of binomial coefficients. displaced Fibonomial a r r a y . We shall also generalize to a The case for a r b i t r a r y rectangular a r r a y s of extended coeffi- cients is treated. 355 356 A NEW PRIMALITY CRITERION OF MANN AND SHANKS 2. [Oct. THEOREMS OF HERMITE According to Dickson's History [2, p. 272] Hermite stated that (2.1) ^ (m and <"> where Srri7Hr|(n) (a,b) denotes the greatest common divisor of a Catalan, Mathews, and Woodall according to Dickson. and b. Proofs were given by What is m o r e , there are extensions to multinomial coefficients and Ricci [5] noted that (2.3) L -pr-j --7 = 0 a. D. • • • c. mod T r—; ? , where a + b + ••• + c = m . ^a, D, * * , c) These theorems have been used in many ways to derive results in number theory. For example, Eq. (2.1) gives us at once that 0 for each k with 1 ^ k ^ p - 1 provided that p is a p r i m e . proofs of Wilson T s criterion and F e r m a t ' s congruence. n + 1 i. e. , the numbers generated by 1, 1, 2, 5, 14, 42, 132, 429, 1430, ••• This has been used in various F r o m (2.2) we obtain at once that m are all integers. This sequence, the so-called Catalan sequence, is of considerable importance in combinatorics and graph theory, and the r e a d e r may consult the historical note of Brown [l] for details. F o r the sake of completeness we wish to include proofs of (2.1) and (2.2) to show how easily they follow from the Euclidean algorithm. By the Euclidean algorithm we know that there exist integers A and B such that (m,n) = d = mA + nB . 1972] A NEW PRIMALITY CRITERION OF MANN AND SHANKS 357 Therefore ri (m - l)(m - 2) ••• fan - n + 1) /m\ A , /m - l\ _ _ so that, upon multiplication by m we have mE , • ( - ) from which it is obvious that f|(») as stated in (2.1). This is essentially Hermite f s proof [3, 415-416, Letter of 17 April 1889], Similarly, there exist integers C and D such that (m + l , n ) = d = (m + 1)C + nD . Rearranging, d = (m - n + 1)C + (C + D)n . Therefore , m(m - 1) * • • (m -£ J ^ L = W c + ^ ^ C + D j c ^ F . a n l n t e g e r , whence on multiplying by m - n + 1 we have (m - n + 1)F , so that m n + 1 I (m\ I \[n) as stated in (2.2). Hermite T s proof may be applied to other similar theorems. The usefulness of Hermite ! s theorems suggested to me that one might get p a r t of the results of Mann and Shanks from them. 3. FIRST RESTATEMENT OF THE CRITERION Leaving the a r r a y arranged as before, it is easy to see that we may restate the condition of Mann and Shanks in the form 358 A NEW PRIMALITY CRITERION OF MANN AND SHANKS (3.1) k = prime if and only if n [Oct. ( n ) \k - 2ny for all integers n such that k / 3 ^ n ^ k / 2 . If we take all entries in the a r r a y other than the binomial coefficients to be z e r o , we can say n for all n. M By Hermite's theorem (2.1) we have in general (n,k - 2n) for all integers 2n ^ k ^ 3n. I (k - 2n) Equivalently, (n,k - 2n)(. " ) = nE , ^ k - 2n) for some integer E. Let k = p r i m e . Then, except for n = k, nj'k, whence njte - 2n. But this means that njf(n, k - 2n), so that n must be a factor of \ k - 2n J The case n = k offers no difficulty since we are only interested in k / 3 ^ n ^ k / 2 . The converse, that k must be prime if n is a factor of n ( ) for every n can be handled along the lines of [4]; ( n 1 ) y k - 2n J when 2|k or 3|k so one must consider p r i m e columns that occur for k of form 6j + 1. 4. SECOND RESTATEMENT OF THE CRITERION Arrange the binomial coefficients in the usual array; k = 0, 1, • • • , n on the n 1 1 1 1 1 1 1 1 3 3 4 6 5 6 1 2 10 15 1 4 10 20 1 5 15 1 6 1 row: 1972] A NEW PRIMALITY CRITERION OF MANN AND SHANKS 359 It is easy to see that the criterion may be stated as follows: (4.1) 2n + 1 = prime if and only if n - k f f _ ~ . 1 I \^2k + i y for every k = 0, l f Since 2n ^ prime for n ^ 1 we may ignore the case whether »-H("i k )Again, Hermite ! s theorem (2.1) is the clue for the proof that n - k is a factor when 2n + 1 is a p r i m e . F o r (2.1) gives us in general n - k (n - k5 2k for all integers 0 ^ k ^ — 5 — . J ^2k + i y TTj Equivalently ? n (n - k, 2k + D (( 22k k '++ k1l j) for some integer 2(n - k), i. e. , E. Suppose n - k j 2n + 1, 2n + 1 = p r i m e . == ((nn * k)E ' If n - k | 2k + 1, then n - k.J 2k + 1 + which is again impossible so that n - k^(n - k, 2k + 1), whence we must have n 5. " k aS 2k + l ) desired " QUESTIONS It would be of interest to see whether (2,2) implies any similar r e s u l t s . We find in gene r a l that and /ro\ (5 2) ' n - 3k j / n - k \ (n - k i- i, 2k *1> I ( 2 k + i j In (5.1) let n - 3k j (n - k + 1, 2k + 1 ) . But n - 3 k | n - k + l - f0r n ^ ° * i ^- k n - 1 * -S- • Then n - 3k | n - k + 1 and n - 3k | 2k + 1. implies also n - 3k j n - k + 1 - (n - 3k) o r again n - 3k J 2k + 1. 360 A NEW PRIMALITY CRITERION OF MANN AND SHANKS [Oct. If then 2k + 1 = prime we again find that n - 3k must divide the binomial coefficient, and this gives us (5.3) 2k + 1 = prime implies n - 3k ( " (2k V*l )1 f01 all for n ^ 5k + 1 . It is easy to find examples of composite 2k + 1 such that n - 3k is not a factor of the binomial coefficient; e.g. , take k = 7 and n = 24: vT(s) Other possibilities suggest themselves. Letting n - 3k | (n - k + 1, 2k + 1) gives again n - 3 k | n - k + l whence also n - 3k j 3(n - k + 1) - (n - 3k) o r n - 3k | 2n + 3. If we then take 2n + 3 = prime we again obtain a useful theorem. It seems clear from just these samples that the theorems of Hermite can suggest quite a variety of divisibility theorems, some of which may lead to c r i t e r i a similar to that of Mann and Shanks. Whether any of these have any strikingly simple forms remains to be seen. One possibility suggested by (2.2) is that /n + k - l \ (n + k,k) f" k£ " I = nE V for some integer E, / from which we may argue that under certain hypotheses n divides ("i-1) Result (5.3) is related to a theorem of Catalan [2, p. 265] to the effect that ml n! J (m + n - 1): provided (m,n) = 1. Finally we wish to recall that some of the results here a r e related to a problem posed by Erdos (with published solution by F. Herzog) [6] to the effect that for every k there exist infinitely many n such that Catalan sequence). (2n)!/nl(n + k)i is an integer (the case k = 1 yields the In fact Erdos claimed a proof that the set of values of n such that this ratio is not an integer has density z e r o . 6. FIBONOMIAL TRIANGLE It is tempting to try and extend the primality criterion to a r r a y s other than the binomial. Consider the a r r a y of Fibonomial coefficients ofHoggatt [7] displaced in the same manner as the a r r a y of Mann and Shanks: 1972] A NEW PRIMALITY CRITERION OF MANN AND SHANKS 0 1 2 0 3 4 5 6 1 1 7 8 9 10 11 12 6 3 1 1 5 15 13 14 15 16 361 17 18 19 1 1 1 1 1 1 2 1 2 3 2' 1 1 3 5 8 1 13 15 5 1 8 40 60 40 8 1 1 13 104 260 260 104 1 21 273 1092 1 34 21 34 Here the Fibonomial coefficients are defined by F F n n-1 F F *k*k-l where F ,- = F n-KL n +F •n-k+1 = 1 - , with F 0 = 0, F 4 = 1, being the Fibonacci numbers. In the d i s - n-1 placed a r r a y , the row numbers a r e made to be the corresponding Fibonacci numbers. From the above sampling, as well as from extended tables, one is tempted to conjecture that a column number is prime if and only if each Fibonomial coefficient in the column is divisible by the corresponding row Fibonacci number. (6.1) k = prime if and only if In other words, it appears that F k - 2n for all k / 3 < n < k/2 . Now as a m a t t e r of fact the exact analogue of Hermite's (2.1) is true: (6.2) w: "m u 1n The proof is an exact replica of Hermite' s proof. What is m o r e , Eq. (2.1) holds true for perfectly a r b i t r a r y a r r a y s in the sense defined in [8], That i s , take an a r b i t r a r y sequence of integers A such that A0 = 0 and A n f 0 for n ^ 1, and define generalized binomial coefficients by (6.3) In/ 1M _ n n-1 n-k+1 A Vk-i- i If all of these turn out to be integers, then we have with h = i 362 A NEW PRIMALITY CRITERION OF MANN AND SHANKS [Oct. A (6.4) (ATA) m' n However, the corresponding form of (2.2) is in general false. The reason is that the step (A § 1 , A ) = d = CA ^ + DA = C(A ^ - A ) + (D + C)A m+1' n m+1 n m+1 n n can be applied only if also m+1 o r something close. n m-n+1 Thus it is not at once clear that the Fibonomial Catalan numbers | 2n | n F a r e integers. n+1 The first few of these are in fact 1, 1, 3, 20, 364, 17017, etc. However, having (2.1) extended to (6.4) is a good result because in o r d e r to obtain a theorem such as (6.5) k = prime implies A k _ 2 v for all k / 3 < n < k/2 it is only necessary to be able to prove that (6.6) (A , A k _ 2 n ) = 1 when k = p r i m e , and k / 3 < n < k/2 . F o r the Fibonacci numbers this is easy because of a known fact that in general (6 7) - <V V Thus we have (F , F, _ 2 ) = F / , 2 = F(a,b) • v so that for k = prime (n, k - 2n) = 1, and since FA = 1 we have the desired result. we know as in (3.1) that The Fibonomial displaced a r r a y criterion then parallels the ordinary binomial case studied by Mann and Shanks. Quite a few standard number-theoretic results have analogues in the Fibonomial case. 7. FIBONOMIAL ANALOGUE OF HERMITE'S SECOND THEOREM Although we have just indicated that divisibility theorem (2.2) does not hold in general for the generalized binomial coefficients, we now show that it does hold for the Fibonomial c o efficients. Thus we now prove that 1972] A NEW PRIMALITY CRITERION OF MANN AND SHANKS 363 F " m-n+1 (7.1) (F m+l' V We need to observe that (a - b,b) = (a,b). This follows, e. g. , from an easily proved stronger assertion that (a + c,b) = (a,b) when b|c. proof given by Hermite for (2.2), as follows. Let (F This lemma is used to modify the - , F ) = d. Now, by (6.7) and our lemma, d = (F m+l' V = for some integers x , y . F m F (m+l,n) = F (m-n+l,n) = < F m-n + l> F n> = xF m-n+l + ^ Therefore F m -n1 m-n+2 F F n . . . F1 = n n-1 1 and by multiplication with F x Jm. <n , + y / n - l = E, some integer, - we obtain E-F m-n+1 •'« i»l • whence F ,, / d is indeed a factor m-n+1 A s a valuable corollary we get the fact that the Fibonomial Catalan numbers a r e inte- gers; this follows at once from (7.1) by setting m = 2n, and noting that (F - , F ) = 1, so that we have (7.2) n+1 The Fibonomial Catalan sequence 1, 1, 3, 20, 364, 17017, 4194036, ••• generated by 2n UL n+l is the exact analogue of the familiar Catalan sequence 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ••• generated by (n + 1) A brief history of the Catalan sequence is given in [1]. A nice proof of (6.7) is given in [9]. 364 A NEW PRIMALITY CRITERION OF MANN AND SHANKS Oct. 1972 8. SOME CONGRUENCES FOR PRIME FIBONACCI NUMBERS It is not known whether there exist infinitely many Fibonacci numbers which a r e p r i m e s , but we may easily obtain congruences which such p r i m e s satisfy. Let F be a prime number. m (6.2) we obtain: Then (F (8.1) then F J ^ l f o r l ^ n ^ m - l . If F m = p r i m e , m , F ) = 1 for n 1 ^ n ^ m - 1. Thus from This is a Fibonacci analogue of the fact that PI ( E j for 1 - k ^ p - 1 if p is a prime. Now, it is known [10] that (8.2) £ (_1)k(k+l)/2 k=0 j m F --l_k = 0> m ^ 2 . ' This & generalizes such relations as F ,- - F - F . = 0, F 2 - 2F 2 - 2F 2 . + F 2 n = 0, a+1 a a-1 a+1 a a-1 a-2 ' etc. See also Hoggatt [7]. Brother Alfred gives a useful table [10] of the Fibonomial coefficients up to m = 12. Identity (8.2) may be used with (8.1) to obtain an interesting congruence; for every t e r m in (8.2) is divisible by F last terms. (8.3) when F If F m = p r i m e , then F m _ 1 = _ ( _ 1 ) m ( m + 1 ) / 2 p 1 * 1 " 1 r » a a_m for all integers a. A special case is of interest. (8.4) is a p r i m e , except the first and Thus we obtain the congruence: If F m = p r i m e , (mod F m ) " > m 2 , » Let a = m + 1 and we get: then F™~* = _ ( _ D m ( m + 1 ) / 2 ( m o d F^)f > m 2 . Another result useful for deriving congruences is the identity 2m+l where the L ' s are Lucas numbers, defined by L 0 = 2, Lj = 1, and L - = L + L _-. The identity was noted in Problem H-63 of Jerbic [11]. If we apply the extended Hermite theorem (8.1) to this, we obtain: (8.6) If F 2 m + 1 = p r i m e , [Continued on page 372. ] then m T T L £ k = 2 (mod F 2 m + 1 ) k=0 .